What is the speed limit of martensitic transformations?

ABSTRACT Structural martensitic transformations enable various applications, which range from high stroke actuation and sensing to energy efficient magnetocaloric refrigeration and thermomagnetic energy harvesting. All these emerging applications benefit from a fast transformation, but up to now their speed limit has not been explored. Here, we demonstrate that a thermoelastic martensite to austenite transformation can be completed within 10 ns. We heat epitaxial Ni-Mn-Ga films with a nanosecond laser pulse and use synchrotron diffraction to probe the influence of initial temperature and overheating on transformation rate and ratio. We demonstrate that an increase in thermal energy drives this transformation faster. Though the observed speed limit of 2.5 × 1027 (Js)1 per unit cell leaves plenty of room for further acceleration of applications, our analysis reveals that the practical limit will be the energy required for switching. Thus, martensitic transformations obey similar speed limits as in microelectronics, as expressed by the Margolus – Levitin theorem.


2) Estimated radiation transmission rate
To estimate if the synchrotron radiation probes the complete film thickness, we calculated the ratio of transmitted X-ray for our film. 1 0 = − I0 and I1 are the X-ray intensities before and after the film, μ is the linear attenuation coefficient of the material and x the distance travelled in the film. To give information about the whole film, the radiation has to travel through the film to the substrate and back out again. As the radiation hits the film in an angle of 18° and with a film thickness of 500 nm, x equals 3200 nm. With μ of 887 cm -1 (for Ni2MnGa with a density of 8 g cm -3 and a beam energy of 12.1 keV, [38]), 1 0 calculates to 75 %, which means that still a reasonable amount of the x-ray radiation is not absorbed in the film. Therefore, we can assume that the synchrotron radiation gives information from the complete film thickness.

3) Characterization of the temperature dependent phase transition
To select reasonable base temperatures for our pump-probe experiments, a thorough understanding of the transition temperatures of the material is necessary. Therefore, the phase transition of the investigated Ni-Mn-Ga thin film was studied using magnetization as well as resistivity measurements, in addition to the diffraction experiments. Fig. S2 (a) shows the temperature-dependent magnetization of the sample, which was measured by a vibrating sample magnetometer (VSM) in a PPMS system (VersaLab TM , 2 K/min heating and cooling rate in an external magnetic field of 0.01 T). The phase transition shows a hysteresis and is accompanied by a change in magnetization at around 350 K. As the transition partly overlaps with the Curie-temperature at 366 K, an additional resistivity measurement was performed in another PPMS system. The resulting curve in zero field is shown in Fig. S2 (b) in black and for clarity only the reverse transition is plotted. As the austenite has a lower resistivity than the martensite, there is a clear drop at the phase transition. For comparison, the phase fraction of the martensite was also measured at the beamline using the diffracted intensities of the (16 -2 0)MM martensite peak without laser heating. The red curve in Fig. S2 (b) shows this phase fraction for particular temperatures, which were held steady within ±0.3 K for the measurement time.
To obtain the martensite phase fraction, the intensity was normalized to the maximum intensity at 306 K. Both curves show a similar transition region, which is shifted to slightly lower temperatures for the curve derived from the diffraction experiments. This may be attributed to the different measurement setups, as the thermometers are placed at different distances to the sample. Furthermore, during the resistivity measurement, the temperature was swept with a rate of 3 K/min compared to the quasi-static diffraction experiments. To determine the austenite start and finish temperatures, we thus used the curve obtained by X-ray diffraction as the corresponding temperature measurement setup was the same one used for all time-dependent measurements at the synchrotron. As with the time dependent measurements, we defined the austenite start temperature as the point were 5 % of the sample has transformed and the finish temperature for 95 % transformation, which gives 336 K and 365 K, respectively. An interesting point that can only be seen from the diffraction experiment (red curve) is that even at 400 Kconsiderably above the sharp intensity dropthe transition to the austenite is not fully completed yet. This aspect is only visible in the direct diffraction experiments and not by the commonly used indirect methods like magnetization and resistivity. Determining transition temperatures in quasi-static measurements. (a) Temperature dependent magnetization of the investigated sample measured with VSM in a PPMS system (VersaLab TM ). An external field of 0.01 T was applied. As the phase transition partially overlaps with the Curie-temperature (Tc = 366 K), an additional resistivity measurement was performed in another PPMS system in zero field ((b), black curve). For comparison, the temperature-dependent intensity of the martensite phase measured quasi-statically at the beamline is shown in red in vicinity of the transformation temperature. Both curves in (b) only depict the reverse transformation. All measurements reveal a similar transformation behavior and slight differences in temperature can be attributed to the different setup and sweep rates. Therefore, we used the values obtained directly at the beamline: austenite start AS= 336 K at 95 % martensite fraction and austenite finish AF= 365 K at 5 %, as marked in the graph.

4) Determination of the laser induced temperature rise
For the detailed evaluation of the results gathered from the time-dependent diffraction experiments, it is necessary to probe the temperature rise ΔT induced by the laser. To obtain this, we use the high accuracy of diffraction to probe lattice parameters, which allows to measure the thermal expansion. Thus, the lattice constant of the Ni-Mn-Ga film itself is the "thermometer", and accordingly ΔT is representative of the relevant sample region and thickness. The calculation of the X-ray radiation transmitted for the present film (section 2) of the supplementary) reveals a diffraction efficiency as high as 75 %, and accordingly we consider these measurements a good average over the film thickness.
For these measurements we heated the sample to 413 K into the fully austenite state to avoid any possible variant reorientation within the martensitic state. We chose the maximum laser fluence of around 60 mJ cm -2 and monitored the position of the austenite (004) peak perpendicular to the substrate. From that, we derived the change in lattice parameter (Δa0) in relation to the initial value before the laser pulse hit the sample, which is plotted in Fig. S3. Figure S3: Determining the laser induced temperature rise. Time-dependent change of lattice parameter of the austenite (a0) measured while heating the sample with a laser fluence of 60 mJ cm -2 from a starting temperature of 413 K. Respective temperatures obtained from a calibration at quasistatic temperatures are shown on the right y-axis. The gray data points were collected with constant diffractometer tilt angles. The red data points show some discrete values with diffractometer angles adjusted for the maximum intensity of the (004)A austenite peak at selected points in time, as described in detail within the text. The blue line shows the calculated temperature development.
The lattice parameter of the film increases sharply while the sample is heated up and decreases afterwards while the sample "slowly" cools down. The black points were With a latent heat of around 5000 J kg -1 [39,40], and a specific heat capacity of around 500 J kg -1 K -1 [26,27], around 10 K of the ΔT will be absorbed by the material as latent heat. We compared the measured values with calculations using the thermal and optical properties of Ni-Mn-Ga (blue line in Figure S3). The maximum temperature change of ΔT* = 187 K is confirmed by the calculation, described in detail in the next section.
However, according to the calculation, the temperature decreases faster after the laser pulse than in the experiment. We attribute this to the film-substrate interface that acts as a heat transfer coefficient, which is not taken into account in the calculation.

5) Calculation of temperature profile during and after laser pulse
To calculate the depth dependent temperature of film and substrate we use the pythonbased toolbox udkm1Dsim [41,42]. This toolbox uses a simplified molecular dynamics model to calculate one-dimensional propagation of coherent and incoherent acoustic phonons. A number of previous measurements showed that at these short laser pulses and large spot sizes heating and subsequent cooling occurs exclusively in the direction perpendicular to the sample surface, i.e., a one-dimensional modelling is sufficient [23].
The simulated structure consists of 1000 unit cells of Ni-Mn-Ga (585 nm) and 30000 unit cells of MgO substrate (12.63 µm). The large thickness of the substrate allows employing a constant temperature boundary condition at the rear side of the substrate. At the sample surface we employ a thermal isolation boundary condition. The complex index of refraction for Ni2MnGa was taken from literature [43] and slightly adapted to match the observed peak temperature rise in the Ni-Mn-Ga film after optical excitation. The value employed in the simulations was 2.0+1.32j. Thermal properties of bulk Ni-Mn-Ga and MgO were taken from [43] and [44], respectively. The optical excitation pulse is absorbed in the upper 65 nm Ni-Mn-Ga film. At this pulse duration, heat within the film dissipates already during absorption of the laser pulse into the substrate. Thus, the generated peak temperature is approximately 10 times lower compared to the excitation with a ultrashort (sub 1 ps) optical pulse. Figure S4 (a) displays the temperature as a function of time and film depth obtained from our simulations. The temperature at the middle of the film matches well the mean film temperature (see Figure S4 (b)).

6) Obtaining characteristic values from the time-dependent intensity measurements
As described in the main part of the paper, our thin film sample was investigated for various different laser fluences as well as base temperatures. To compare the timedependent phase transition for these different experimental conditions, we first had to extract some characteristic parameters from the intensity data. Starting with the impact of the laser pulse, the intensity curves for the martensite to austenite transition are Z-shaped as more and more of the sample transforms until a maximum of transformed phase is reached. To derive characteristic values from these curves, we fitted the part of the curve in which the intensity decreases using a generalized logistic function. This function can be used to describe saturation and transition processes and requires only a small number of fitting parameters. Three exemplary curves with the corresponding fit (red) are shown in Figure S5 (a-c). The generalized logistic function to describe the time dependency of intensity I(t) has theform: (1 + − ⋅( − 0 ) ) 1 UB refers to the upper boundary of the function, i.e., the maximum it approaches. LB is the lower boundary, t0 shifts the position along the time axis, and the parameters b and v describe the shape of the drop. b is the transition rate and v affects how the asymptote is approached. After fitting this function to our data, we derived the following key parameters of a transition process: while UB and LB can be taken directly from the fit, we are additionally interested in the transition time Δt, which is sketched in blue in Figure   S5 (a). We define Δt as time span required for the fit function to decrease from 95 % to 5 % of its maximum drop (UB − LB). This approach gives the transition time, which is the key parameter to understand the speed of a martensitic transformation. In particular, this approach allows neglecting details of the curve shape, which may be affected e, g. by the inhomogeneous temperature profile, described in supplementary section 6). We use the generalized logistic function, as it directly give UB and LB and a slightly better fit to our data compared to the standard one. Together with Δt these data could also be extracted directly from our measurements by the classical tangent method, but with lower accuracy.
For completeness, all fit parameters are summarized in the following table. A generalized logistic function can describe the data points quite well. For the measurement shown in (a), we used a small time increment describing the intensity drop very detailed. Due to the limited measurement time at a synchrotron, we had to increase the step size for most of the experiments as can be seen in (b) and (c). The fit is still able to reproduce the data, which is also the case when the sample only partially transforms as shown in Figure S5 (c). Depending on the experimental conditions, the transition time differs noticeable from well below 10 ns in (b) to nearly 20 ns in (c).

Fig. S5:
Analyzing time-dependent measurements while heating the sample with a nanosecond laser pulse (exemplary measurements). The intensity of the martensite (16 -2 0)MM peak (black) was fitted with a generalized logistic function (red), as described in the text. The upper (UB) as well as lower boundary (LB) of the function and the derived transition times (Δt, sketched in blue) are given in the figure. The base temperature before the laser pulse (T0) was 312 K (a), 336 K (b) and 329 K (c). The laser fluence (E) and the temperature rise ΔT is given in the corresponding graphs as well. As described in supplementary section 4), ΔT was corrected for the latent heat during the transition (*) by subtracting 10 K from the values obtained from the measurements shown in Fig. S3.

7) Estimation of film stress and influence on transformation temperature
In our analysis we focus on the influence of temperature, but the difference of thermal expansion of the hot, thin film on the cold, thick substrate can result in a film stress, which commonly increases the martensitic transformation temperature. Following our analysis of thermal stress in Ni-Mn-Ga films [45] we estimate the increase of transition temperature by thermal stress for the maximum temperature rise of 177 K. From the thermal expansion coefficient of 22.4 x 10 -6 K -1 of Ni-Mn-Ga (see section 4) of this supplementary) we expect a strain of 0,4 %, which is equivalent to a stress of 120 MPa when using the E-module = 20 GPa of austenite [46]. Following the Clausius-Claperyon equation of Ni-Mn-Ga [47,48], this stress increases the transition temperature by 54 K. This, however, is an upper estimate, as this stress is compressive, and films after growth often exhibit tensile stress [45], which can compensate most of the thermal stress. When using the tensile stress of this paper as an approximation (the stress of the present film is not known), the increase of transition temperature is reduced to 9 K. When putting these values in relation to the temperature rise of 177 K, we underestimate the driving energy by 5 % to 30 % by neglecting thermal stress.
We would like to add, that at no time the surface temperature of our film (Fig. S4a) approaches the melting point and thus no shock wave by evaporating forms, as used in other dedicated setups [49]. SEM micrographs of the examined Ni-Mn-Ga film in different magnification. The martensitic microstructure consists mostly of Type Y and the mesoscopic twin boundaries are 20…50 nm apart. We would like to add that we recently published a comprehensive explanation how and why this hierarchical microstructure forms (after slow cooling) [22].